A Finiteness Theorem for Special Unitary Groups of Quaternionic Skew-Hermitian Forms with Good Reduction

摘要

Given a field (K) equipped with a set of discrete valuations (V), we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion (K)-algebra (Q) to quadratic forms over the function field (K(Q)) obtained via Morita equivalence. Using this we show that if ((K,V)) satisfies certain conditions, then the number of (K)-isomorphism classes of the universal coverings of the special unitary groups of quaternionic skew-hermitian forms that have good reduction at all valuations in (V) is finite and bounded by a value that depends on size of a quotient of the Picard group of (V) and the size of the kernel and cokernel of residue maps in Galois cohomology of (K) with finite coefficients. As a corollary we prove a conjecture of Chernousov, Rapinchuk, Rapinchuk for groups of this type.